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ORIGINAL ARTICLE
Analytical solution for the pull-out response of FRP rodsembedded in steel tubes filled with cement grout
Zhimin Wu Æ Shutong Yang Æ Jianjun Zheng ÆXiaozhi Hu
Received: 4 June 2008 / Accepted: 19 June 2009 / Published online: 1 July 2009
� The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract Fiber-reinforced plastic (FRP) tendons
have been widely used for ground anchors in civil
engineering. Although various pull-out tests of FRP
rods from grout-filled steel tubes have been conducted
to simulate ground anchors in rock, there are relatively
few theoretical studies reported in the literature for this
type of bonded anchorages. The intention of this paper
is to present an analytical solution for predicting the
maximum pull-out load of FRP rods embedded in steel
tubes filled with cement grout. First, the expression of
the shear stress along the thickness direction of the
grout layer is obtained analytically. The tensile stress
in the rod and the interfacial shear stress at the rod–
grout interface are formulated at different loading
stages. By modeling interfacial debonding as an
interfacial shear crack, the pull-out load is then
expressed as a function of the interfacial crack length.
Finally, based on the Lagrange multiplier method, the
maximum pull-out load and the critical crack length
are determined. The validity of the proposed analytical
solution is verified with the experimental results
obtained from literature. It can be concluded that the
proposed analytical solution can predict the maximum
pull-out load of spiral wound and indented rods
embedded in steel tubes filled with cement grout with
reasonable accuracy. The proposed solution can be
also applied in predicting the pull-out capacity of steel
bars from concrete.
Keywords FRP rod � Cement grout �Steel tube � Anchorage � Maximum pull-out load
1 Introduction
Fiber-reinforced plastic (FRP) rods are regarded as a
better alternative to steel bars due to their high
strength-to-weight ratio, resistance to corrosion, and
ease of transportation and handling. Integrating fiber
optic sensors into the FRP rods allows us to monitor
Z. Wu � S. Yang
State Key Laboratory of Coastal and Offshore
Engineering,
Dalian University of Technology, Dalian 116024,
People’s Republic of China
Z. Wu
e-mail: [email protected]
S. Yang (&)
School of Civil Engineering and Architecture,
Shandong University of Science and Technology,
Qingdao 266510, People’s Republic of China
e-mail: [email protected]
J. Zheng
School of Civil Engineering and Architecture,
Zhejiang University of Technology, Hangzhou 310014,
People’s Republic of China
e-mail: [email protected]
X. Hu
Department of Mechanical and Materials Engineering,
University of Western Australia, Nedlands, Perth,
WA 6907, Australia
e-mail: [email protected]
Materials and Structures (2010) 43:597–609
DOI 10.1617/s11527-009-9515-x
the behavior of anchors satisfactorily [33]. As the
FRP rod is widely used in ground anchorages such as
grouted anchors embedded in rock, the bond proper-
ties between the FRP rod and cement grout have
gradually attracted more attention of researchers and
engineers.
Erki and Rizkalla [9, 10] introduced detailed
anchorages for glass FRP (GFRP) and carbon FRP
(CFRP) tendons, aramid fiber ropes and aramid FRP
(AFRP) rods giving due considerations of their char-
acteristics. These types of anchorages deserve more
attention in practical applications due to the high axial-
to-lateral strength of FRP materials [22]. The bonded
anchorage is one of the currently-used methods for
FRP rods, in which a FRP rod is bonded into a steel pipe
or tube filled with cement or resin grout [32]. The steel
pipe or tube acts as an infinite rock mass [30]. Pull-out
tests of FRP rods from grout-filled steel tubes are
generally carried out to simulate the ground anchors in
rock [5, 7, 22, 27, 30, 32, 33]. Mckay and Erki [22]
found that the performance of cement grouted anchors
depends on the confinement, moist curing and stiffness
properties of the grout. Budelmann et al. [7] investi-
gated the fatigue behavior of FRP bars anchored in a
cylindrical steel tube filled with a quartz sand mortar.
The introduction of sand and swelling agent into grout
can create pressure on the rod and therefore increase
the shear bonding resistance [5, 22, 23]. On the other
hand, the shrinkage of cement grout decreases the shear
strength [5, 32]. The load-bearing capacity of the
anchorage increases as the bonding length and com-
pressive strength of cement grout, and the elastic
modulus of the steel sleeve increase [3, 5, 8]. If the steel
sleeve too large, however, the effect of its stiffness
becomes less significant [2]. Different surface geom-
etries and mechanical properties of FRP rods can yield
different bonding resistances and pull-out properties
[6, 30]. Moreover, the multi-grouted anchor has been
recommended for practical engineering applications
due to its higher stiffness and load-bearing capacity
than the single grouted anchor [33]. Zhang et al. [34]
carried out a test on a full-scale ground anchor with
fiber-reinforced polymeric 9-bar tendons and found
that the tendons perform satisfactorily in post-tension-
ing applications. Benmokrane et al. [4] replaced steel
tubes with concrete in laboratory tests and with rock in
field tests as host media for pull-out tests. Results
showed that the bond strength from the laboratory tests
is higher than that from the field tests.
Although much experimental work has been
carried out on grouted FRP bars anchored in
mortar-filled steel tubes, theoretical studies are rela-
tively few in the literature. Zhang et al. [32] presented
an analytical model for predicting the tensile capacity
of bonded anchorages. In their model, four parame-
ters related to the FRP rod–grout interface, as well as
the distribution of the bond stress along the embed-
ment length in the ultimate state, are required to write
equilibrium equations. The theoretical results are in
good agreement with the test results.
Since FRP rods are a possible replacement of steel
bars, current studies for pull-out of metallic bars from
cementitious matrix can provide some references on
the bonding characteristics between the FRP rod and
cement grout. The interaction between the bar and
concrete is generally characterized by four different
stages [11]. At the first stage, bond efficiency is
assured by chemical adhesion without any slip. Then
the chemical adhesion breaks down, and interfacial
slip occurs. The bond strength and stiffness are assured
mostly by the interlocking among the reinforcements
at the following stage. Finally, splitting-induced pull-
out failure or single pull-out failure occurs depending
on the transverse confinement or the thickness of the
concrete cover. Li et al. [20] introduced two methods
to improve the bond properties by modifying the
matrix and the rebar surface. Results showed that both
methods can (a) improve the bond performance, (b)
increase the interfacial microhardness and (c) reduce
the porous region of the interface. Bamonte et al. [1]
investigated the size effect of bonding of anchorages to
ordinary and high performance concrete, and found
that the bond is less size-dependent in high perfor-
mance concrete due to the greater tensile strength,
homogeneity and chemical adhesion with the bar.
Moreover, the models for pull-out of fibers from
matrix provide some theoretical guidance. A shear-
lag model was frequently applied in theoretical
studies, in which the tensile stresses in the matrix
are negligible compared with those in the fiber [13].
Based on this model, the behavior of the composites
before and after debonding was extensively analyzed
[14–19, 29]. The fracture energy-based criterion [12,
17, 19, 24, 28, 29] was used to describe the interfacial
debonding behavior. The advantages of this criterion
were detailedly discussed by Stang et al. [29]. Zhang
et al. [31] presented an improved model to obtain the
stress fields in both bonded and debonded regions at
598 Materials and Structures (2010) 43:597–609
the fiber–matrix interface by considering a pull-out
rate-dependent frictional coefficient. The whole pro-
cess of pulling out a single fiber was then modeled
numerically [21]. Naaman and Namur et al. [25, 26]
accurately obtained the entire pull-out load-end slip
relationship of fibers using a dynamic mechanism, in
which the Poisson’s effect, shrink-fit and fiber–matrix
misfit theory were incorporated.
Based on the pioneer work by Zhang et al. [32], an
analytical solution is proposed in this paper for the
maximum pull-out load of FRP rods embedded in
steel tubes filled with cement grout. One advantage of
the proposed solution is that no assumption is made
on the bond stress distribution along the embedment
length. Finally, the validity of the proposed solution
is verified against some experimental results obtained
from the literature.
2 The maximum pull-out load of FRP rods
2.1 Basic assumptions
According to the setup of a typical pull-out test given
in the previous studies [5, 30], a simplified model is
proposed for a cement-grouted FRP rod anchored in a
steel tube as shown in Fig. 1, where D denotes the rod
diameter, t the grout layer thickness, b the tube
thickness, L the embedment length, P the pull-out
load, and q the uniform force acting on the top
surface of the steel tube. The assumptions used for
the analytical solution are as follows:
a. The cement grout and steel tube are both linear
elastic materials with Young’s moduli Ec and Es,
respectively.
b. The bond between the tube and grout is perfect,
i.e., there is no shear slip, elastic slip or debond-
ing at the interface.
c. The normal stress is uniform on the cross-section
of the steel tube.
d. The grout with shear modulus G is in a state of
pure shear.
e. Based on the experimental results from Zhang and
Benmokrane [30], the relationship between the
shear stress s and slip d at the rod–grout interface is
multi-linear as shown in Fig. 2. The grout sleeve is
modeled as a shear-lag member, whose shear
stiffness k is equal to the slope of the ascending
portion in the s–d curve. Thus, the relationship is
given by
s ¼ kd 0� d� d1ð Þ ð1aÞ
s¼ sud2� ssd1
d2�d1
� su� ss
d2�d1
d d1\d�d2ð Þ ð1bÞ
s ¼ ss d[ d2ð Þ; ð1cÞ
where su and ss are the shear strength and residual
frictional stress at the rod–grout interface,
respectively.
Dt tb b
P
)2( tbDb
Pq
)2( tbDb
Pq
L
0
x
r
D
t tb b
FRP rod
Cement grout
Steel tube
Fig. 1 Stress distribution
and geometrical dimensions
of simplified anchorage
specimen
Materials and Structures (2010) 43:597–609 599
f. All radial effects of the rod, grout and steel tube
are neglected.
g. Since the radial stiffness of the confining
medium (i.e., the steel tube) is relatively large
for this type of anchored FRP rods and the shear
dilation of the grout is neglected, interfacial
debonding at the rod–grout interface is the most
likely failure mode.
2.2 The numerical model
A cylindrical rod element and a cylindrical grout
shell element as shown in Fig. 3 are considered,
where rp is the tensile stress in the rod and sr is the
shear stress in the grout layer at the distance r from
the x axis. According to the principle of equilibrium
of forces shown in Fig. 3a, s can be expressed in
terms of rp as
s ¼ D
4
drp
dx: ð2Þ
For the grout element shown in Fig. 3b, equilibrium
yields
1
srdsr ¼ �
1
rdr: ð3Þ
Solving Eq. 3, we have
sr ¼D
2rs: ð4Þ
If u denotes the longitudinal displacement in the
grout, sr can be expressed as
sr ¼ �Gdu
dr: ð5Þ
Substituting Eq. 4 into Eq. 5 and integrating u with
respect to r, we have
u ¼ um þDs2G
lnD
2r; ð6Þ
where um is the longitudinal displacement at the rod–
grout interface. According to assumption (b), the
longitudinal displacement us of the steel tube is equal
to u at r = t ? D/2, i.e.,
us ¼ um þDs2G
lnD
Dþ 2t: ð7Þ
If up denotes the longitudinal displacement of the rod,
the interfacial slip at the rod–grout interface is equal to
0
s
u
1 2
k
1
Fig. 2 Relationship between interfacial shear stress and slip
D
p
pp d
dx r r dx
x axis
r rdr dr
rr d rr d
(a) (b) Fig. 3 Stress analysis of
rod and grout elements
600 Materials and Structures (2010) 43:597–609
d ¼ up � um: ð8Þ
When the rod–grout interface is in an elastic state, scan be obtained by combining Eq. 1a with Eqs. 7 and
8 as follows
s ¼ 2kG
2Gþ Dk ln Dþ2tD
ðup � usÞ: ð9Þ
Differentiating s with respect to x yields
dsdx¼ 2kG
2Gþ Dk ln Dþ2tD
rp
Ep� rs
Es
� �; ð10Þ
where Ep is the Young’s modulus of the FRP rod and
rs is the normal stress in the steel tube. According to
the equilibrium of forces on the cross-section shown
in Fig. 4, the relationship between rp and rs is
rs ¼ �D2
4bðDþ bþ 2tÞ rp: ð11Þ
Substitution of Eqs. 2 and 11 into Eq. 10 yields
d2rp
dx2� a2rp ¼ 0; ð12Þ
where
a2 ¼ 8kG
D 2Gþ Dk ln Dþ2tD
� � 1
Epþ D2
4EsbðDþ bþ 2tÞ
� �:
ð13Þ
By introducing the following boundary conditions
rpðx ¼ 0Þ ¼ 0 ð14Þ
rpðx ¼ LÞ ¼ 4P
pD2ð15Þ
rp and s have the following expressions:
rp ¼4P
pD2
eax � e�ax
eaL � e�aLð16Þ
s ¼ PapD
eax þ e�ax
eaL � e�aL: ð17Þ
When P is equal to the initial cracking load Pini, s at
the loaded end (i.e., x = L) reaches the interfacial
shear strength su, i.e.,
Pini ¼supD
aeaL � e�aL
eaL þ e�aL: ð18Þ
When P continues to increase, an interfacial debond-
ing region, referred to as the interfacial crack
hereafter, will occur near the loaded end. According
to the relationship between s and d shown in Fig. 2,
there is a single interfacial crack, but two interfacial
behaviors, with either softening or friction. Once
P [ Pini, a softening crack of length as first appears at
the loaded end as shown in Fig. 5.
Since the rod–grout interface of 0 B xB L - as is
still in an elastic state, rp and s can be obtained by
solving Eq. 12 and applying the conditions of Eq. 14
and s(x = L - as) = su as follows
ps s
Fig. 4 Normal stresses on
the cross-section of the
anchorage specimen
P
)2( tbDb
Pq
)2( tbDb
Pq
0
x
sa
Softening crack
Fig. 5 Softening crack propagation along the rod–grout
interface
Materials and Structures (2010) 43:597–609 601
rp ¼4su
Daeax � e�ax
eaðL�asÞ þ e�aðL�asÞð19Þ
s ¼ sueax þ e�ax
eaðL�asÞ þ e�aðL�asÞ: ð20Þ
In the region of L - as \ x B L, the interface is in a
softening state. By combining Eq. 1b with Eqs. 7 and
8, s is related to up and us by
s ¼ � 2Gðsu � ssÞ2Gðd2 � d1Þ þ Dðsu � ssÞ ln D
Dþ2t
ðup � usÞ
þ 2Gðsud2 � ssd1Þ2Gðd2 � d1Þ þ Dðsu � ssÞ ln D
Dþ2t
:
ð21Þ
Differentiating s with respect to x yields
dsdx¼ � 2Gðsu � ssÞ
2Gðd2 � d1Þ þ Dðsu � ssÞ ln DDþ2t
rp
Ep� rs
Es
� �:
ð22Þ
Substituting Eqs. 2 and 11 into Eq. 22, we can obtain
another differential equation as follows
d2rp
dx2þ x2rp ¼ 0; ð23Þ
where
x2 ¼ 8Gðsu � ssÞ2DGðd2 � d1Þ þ D2ðsu � ssÞ ln D
Dþ2t
1
Epþ D2
4EsbðDþ bþ 2tÞ
� �:
ð24Þ
By solving Eq. 23 and considering the continuity
conditions of rp and s at x = L - as, rp and s can be
determined as follows
rp ¼"
4su
DatanhðaðL� asÞÞ cos xðL� asÞ
� 4su
Dxsin xðL� asÞ
#cos xx
þ"
4su
Da� tanhðaðL� asÞÞ sin xðL� asÞ
þ 4su
Dxcos xðL� asÞ
#sin xx ð25Þ
s ¼ � su
"xa
tanhðaðL� asÞÞ cos xðL� asÞ
� sin xðL� asÞ� sin xx
þ su
"xa� tanhðaðL� asÞÞ sin xðL� asÞ
þ cos xðL� asÞ� cos xx: ð26Þ
By substituting Eq. 15 into Eq. 25, P can be
expressed in terms of as as follows
P ¼ supD
atanhðaðL� asÞÞ cos xas þ
supD
xsin xas:
ð27Þ
When the ultimate state is reached, the critical
softening crack length asc can be determined by
solving dP/das = 0 and the maximum pull-out load
Pmax is then obtained by inserting asc into Eq. 27. The
interfacial shear stress s0 at the free end (i.e., the free
end) can be obtained by inserting x = 0 into Eq. 20
as follows
s0 ¼2su
eaðL�asÞ þ e�aðL�asÞ: ð28aÞ
From Eq. 28a, it can be seen that s0 is always smaller
than su, which shows that no interfacial debonding
will occur at the free end. The interfacial shear stress
sL at the loaded end can be obtained by inserting
x = L and as = asc into Eq. 26 as follows
sL ¼ su cos xasc �suxa
tanhðaðL� ascÞÞ sin xasc:
ð28bÞ
In Eq. 28b, sL is larger than ss. When sL is smaller
than ss, a frictional crack of length af will develop at
the loaded end before the ultimate state occurs as
shown in Fig. 6.
For the region of 0 B x B L - as - af, rp and sare given by solving Eq. 12 and combining with
Eq. 14 and s(x = L - as - af) = su as follows
rp ¼4su
Daeax � e�ax
eaðL�as�af Þ þ e�aðL�as�af Þð29Þ
s ¼ sueax þ e�ax
eaðL�as�af Þ þ e�aðL�as�af Þ: ð30Þ
For the region of L - as - af \ x B L - af, rp and
s are obtained by solving Eq. 23 and considering the
602 Materials and Structures (2010) 43:597–609
continuity conditions of rp and s at x = L - as - af
that
rp ¼"
4su
DatanhðaðL� as � af ÞÞ cos xðL� as � af Þ
� 4su
Dxsin xðL� as � af Þ� cos xx
þ"
4su
DatanhðaðL� as � af ÞÞ sin xðL� as � af Þ
þ 4su
Dxcos xðL� as � af Þ� sin xx ð31Þ
s ¼ � su
"xa
tanhðaðL� as � af ÞÞ cos xðL� as � af Þ
� sin xðL� as � af Þ� sin xx
þ su
"xa
tanhðaðL� as � af ÞÞ sin xðL� as � af Þ
þ cos xðL� as � af Þ� cos xx: ð32Þ
The continuity condition of s(x = L - af) = ss
yields the following equation
cos xas �xa
tanhðaðL� as � af ÞÞ sin xas �ss
su¼ 0:
ð33Þ
For the frictional region of L - af \ x B L, s is
always equal to ss. Thus, another differential equation
can be derived by using Eq. 2 as follows
drp
dx¼ 4ss
D: ð34Þ
Solving Eq. 34 and considering the continuity con-
dition of rp at x = L - af, we have
rp ¼4ss
Dxþ 4su
DatanhðaðL� as � af ÞÞ cos xas
þ 4su
Dxsin xas �
4ss
DðL� af Þ: ð35Þ
By combining the boundary condition of Eq. 15 with
Eq. 35, P can be expressed in terms of as and af as
follows
P ¼ sspDaf þsupD
atanhðaðL� as � af ÞÞ cos xas
þ supD
xsin xas: ð36Þ
To obtain the maximum pull-out load Pmax, the
Lagrange multiplier method is adopted in this paper.
For this purpose, a Lagrange function U(as, af, k) is
constituted from Eqs. 33 and 36 as follows
Uðas; af ; kÞ ¼ sspDaf þsupD
atanhðaðL� as � af ÞÞ
cos xas þsupD
xsin xas
þ k
"cos xas �
xa
tanhðaðL� as � af ÞÞ
sin xas �ss
su
#;
ð37Þ
where k is an unknown parameter to be solved. By
applying the following conditions
oUoas¼ oU
oaf¼ oU
ok¼ 0 ð38Þ
three equations can be established. By solving the three
equations, the critical crack lengths asc and afc, and the
maximum pull-out load Pmax can be determined.
3 Experimental verification and discussion
To verify the proposed analytical solution, some
experimental results obtained from the literature are
selected for comparison [32]. The basic parameters
used in the analytical solution are shown in Table 1. In
the experiment of Zhang et al. [32], three types of FRP
P
)2( tbDb
Pq
)2( tbDb
Pq
0
x
sa
faFrictional crack
Softening crack
Fig. 6 Softening crack and frictional crack propagation along
the rod–grout interface
Materials and Structures (2010) 43:597–609 603
rods, classified as round sanded, spiral wound and
indented, and four types of cement grouts denoted as
CG1, CG2, CG3 and CG4 were prepared and tested.
The mixture proportions of the four grouts are shown in
Table 2. In each test, the specimens with the embed-
ment lengths of 100 mm (D = 7.5, 7.9 and 8.0 mm)
were used to determine the interfacial parameters as
shown in Table 3 [32]. The shear modulus G of grout
for each type of specimens can be calculated by
G ¼ Ec
2ð1þ tÞ : ð39Þ
With these parameters, Pmaxa can be determined
by applying the proposed analytical procedure, to
make comparisons with the experimentally-measured
Pmaxe as shown in Table 4.
It can be seen from Table 4 that, except for Nos. 5
and 6 specimens, the calculated results are in good
agreement with the experimental results. For Nos. 5
and 6 specimens, the surface of FRP rods is round
sanded and their embedment lengths are relatively
long. The bond between the round sanded rod and
grout mainly depends on chemical adhesion and
friction once the interfacial slip occurs [30]. In
addition, since the elastic modulus of the round
sanded rod is relatively low (60.83 GPa), the effect of
the radial shrinkage in the rod becomes more
significant due to the larger maximum pull-out
strains. As a result, the interfacial shear strength su
and residual frictional stress ss decrease, which has
not been taken into account in the proposed analytical
solution. For other specimens, however, the bond
between the spiral wound or indented rod and grout is
mainly due to the interlocking interaction. The
compressive interaction between the spiral or rib
and grout provides the resistance for the rod. Thus,
the radial shrinkage in these rods has a minor effect
on the interfacial shear strength su and residual
frictional stress ss. Therefore, the effect of the radial
shrinkage in FRP rods should further be taken into
Table 1 Basic geometrical and mechanical parameters of anchorage specimens from Zhang et al. [32]
No. Type D (mm) t (mm) b (mm) L (mm) Ec (GPa) Poisson’s ratio tof grout
Ep (GPa) Es (GPa)
1 Round sanded ? CG1 7.5 21.75 3.0 100 17.4 0.11 60.83 195
2 Round sanded ? CG2 7.5 21.75 3.0 100 18.6 0.11 60.83 195
3 Round sanded ? CG3 7.5 21.75 3.0 100 22.9 0.10 60.83 195
4 Round sanded ? CG4 7.5 21.75 3.0 100 16.7 0.12 60.83 195
5 Round sanded ? CG4 7.5 21.75 3.0 200 16.7 0.12 60.83 195
6 Round sanded ? CG4 7.5 21.75 3.0 350 16.7 0.12 60.83 195
7 Spiral wound ? CG1 8.0 21.50 3.0 100 17.4 0.11 43.50 195
8 Spiral wound ? CG2 8.0 21.50 3.0 100 18.6 0.11 43.50 195
9 Spiral wound ? CG3 8.0 21.50 3.0 100 22.9 0.10 43.50 195
10 Spiral wound ? CG4 8.0 21.50 3.0 100 16.7 0.12 43.50 195
11 Spiral wound ? CG4 8.0 21.50 3.0 200 16.7 0.12 43.50 195
12 Spiral wound ? CG4 8.0 21.50 3.0 350 16.7 0.12 43.50 195
13 Indented ? CG1 7.9 21.55 3.0 100 17.4 0.11 163.33 195
14 Indented ? CG2 7.9 21.55 3.0 100 18.6 0.11 163.33 195
15 Indented ? CG3 7.9 21.55 3.0 100 22.9 0.10 163.33 195
16 Indented ? CG4 7.9 21.55 3.0 100 16.7 0.12 163.33 195
17 Indented ? CG4 7.9 21.55 3.0 200 16.7 0.12 163.33 195
Table 2 Mixture proportions of four grouts from Zhang et al.
[32]
No. Mixture proportions
CG1 Type 10 portland cement (ASTM 1)
CG2 Type 30 portland cement (ASTM II) ? superplasticizer
solids (1% by weight of cement)
CG3 Type 10 portland cement (ASTM 1) ? sand (40% by
weight of cement)
CG4 Type SF cement (blended Type I cement containing 8%
silica fume) ? swelling agent (0.004% by weight of
cement and other additives)
604 Materials and Structures (2010) 43:597–609
account in future studies to predict the maximum
pull-out load more accurately.
The effect of the embedment length-to-rod diam-
eter ratio on the distribution of interfacial shear stress
in the ultimate state is shown in Figs. 7, 8 and 9 for
three types of FRP rods, respectively. From Figs. 7 to
9, it can be seen that the smaller the embedment
length-to-rod diameter ratio is, the more uniform is
the interfacial shear stress. When the embedment
length is equal to 100 mm (L/D = 13.3, 12.7 or
12.5), the variation of the interfacial shear stress is
small. In this case, the interfacial shear stress can be
assumed to be uniform as by Zhang et al. [32] and the
corresponding interfacial parameters obtained from
specimens with embedment length 100 mm are
reliable.
To further verify the proposed analytical solution,
the data by Zhang and Benmokrane [30] are selected
Table 3 Interfacial parameters obtained from Zhang et al. [32]
Type No. su (MPa) ss (MPa) d1 (mm) d2 (mm)
Round sanded ? CG1 Specimen 1 8.2 2.8 1.31 3.86
Round sanded ? CG2 Specimen 2 7.9 2.5 1.05 6.10
Round sanded ? CG3 Specimen 3 8.4 3.1 0.72 5.60
Round sanded ? CG4 Specimens 4–6 8.7 2.6 0.66 4.18
Spiral wound ? CG1 Specimen 7 12.3 3.3 2.34 7.66
Spiral wound ? CG2 Specimen 8 7.9 2.4 2.30 6.48
Spiral wound ? CG3 Specimen 9 12.3 3.3 1.78 7.80
Spiral wound ? CG4 Specimens 10–12 13.2 3.8 2.50 6.50
Indented ? CG1 Specimen 13 13.1 4.1 3.32 9.60
Indented ? CG2 Specimen 14 10.6 3.1 2.97 9.95
Indented ? CG3 Specimen 15 12.4 4.4 2.61 8.70
Indented ? CG4 Specimens 16–17 14.4 5.6 2.90 6.40
Table 4 Comparison
between analytical results
and experimental results
from Zhang et al. [32]
Numbers of
specimens
Types of Specimens Pmaxa (kN) Pmax
e (kN) (Pmaxe - Pmax
a )/Pmaxe
9 100 (%)
1 Round sanded ? CG1 18.8 19.4 3.1
2 Round sanded ? CG2 18.3 18.6 1.6
3 Round sanded ? CG3 19.5 19.9 2.0
4 Round sanded ? CG4 20.0 20.6 2.9
5 Round sanded ? CG4 36.9 26.9 -37.2
6 Round sanded ? CG4 51.9 37.1 -39.9
7 Spiral wound ? CG1 30.0 30.9 2.9
8 Spiral wound ? CG2 19.4 20.0 3.0
9 Spiral wound ? CG3 30.0 31.0 3.2
10 Spiral wound ? CG4 31.8 33.3 4.5
11 Spiral wound ? CG4 56.2 55.6 -1.1
12 Spiral wound ? CG4 74.5 67.9 -9.7
13 Indented ? CG1 32.3 32.6 0.9
14 Indented ? CG2 26.2 26.7 1.9
15 Indented ? CG3 30.6 30.8 0.6
16 Indented ? CG4 35.3 35.8 1.4
17 Indented ? CG4 68.0 67.6 -0.6
Materials and Structures (2010) 43:597–609 605
for comparison. The main difference compared with
the first series of data [32] is that the present grout
thickness t is much lower as shown in Table 5. The
adopted specimens are composed of two types of FRP
rods classified as round sanded and ribbed, and two
types of cement grouts denoted as CM and EM as
shown in Table 6. For each test, the interfacial
parameters are selected from the specimens with
embedment length of 40 mm (D = 7.5 and 7.9 mm)
as shown in Table 7.
It can be seen from Table 8 that, except for the No.
SP2 specimen, the calculated results are in good
agreement with the experimental results. For the No.
SP2 specimen, the surface of the FRP rod is round
sanded and its bond mechanism with the grout is
friction-resistance type. Since the elastic modulus of
the rod is relatively low (60.8 GPa), the effect of the
radial shrinkage in the rod becomes more significant
due to the larger maximum pull-out strain as
explained above for Nos. 5 and 6 specimens.
0 50 100 150 200 250 300 3502
3
4
5
6
7
8
9
10
Distance from the free end (mm)
Inte
rfac
ial s
hear
str
ess
(MPa
)
No. 4 Specimen (L/D=13.3, P/Pmax=1, round sanded rods)
No. 5 Specimen
(L/D=26.7, P/Pmax=1, round sanded rods)
No. 6 Specimen (L/D=46.7, P/Pmax=1, round sanded rods)
Fig. 7 Distribution of
interfacial shear stress along
the length of the sand
rounded FRP rod
0 50 100 150 200 250 300 3502
4
6
8
10
12
14
Distance from the free end (mm)
Inte
rfac
ial s
hear
str
ess
(MPa
)
No. 10 Specimen(L/D=12.5, P/Pmax=1, spiral wound rods)
No. 11 Specimen (L/D=25, P/Pmax=1, spiral wound rods)
No. 12 Specimen (L/D=43.75, P/Pmax=1, spiral wound rods)
Fig. 8 Distribution of
interfacial shear stress along
the length of the spiral
wound FRP rod
606 Materials and Structures (2010) 43:597–609
0 50 100 150 200 250 300 3508
9
10
11
12
13
14
15
Distance from the free end (mm)
Inte
rfac
ial s
hear
str
ess
(MPa
)
No. 16 Specimen (L/D=12.7, P/Pmax=1, indented rods)
No. 17 Specimen (L/D=25.3, P/Pmax=1, indented rods)
L/D=44.3, P/Pmax=1, indented rods
Fig. 9 Distribution of
interfacial shear stress along
the length of the indented
FRP rod
Table 5 Basic geometrical and mechanical parameters of anchorage specimens from Zhang and Benmokrane [30]
Numbers of
specimens
Types of specimens D (mm) t (mm) b (mm) L (mm) Ec (GPa) Poisson’s ratio
t of grout
Ep (GPa) Es (GPa)
SP1 Round sanded ? EM 7.5 8.95 4.8 40 22.6 0.24 60.8 195
SP2 Round sanded ? EM 7.5 8.95 4.8 80 22.6 0.24 60.8 195
SP3 Ribbed ? CM 7.9 8.75 4.8 40 26.6 0.22 163.3 195
SP4 Ribbed ? CM 7.9 8.75 4.8 80 26.6 0.22 163.3 195
SP5 Ribbed ? EM 7.9 8.75 4.8 40 22.6 0.24 163.3 195
SP6 Ribbed ? EM 7.9 8.75 4.8 80 22.6 0.24 163.3 195
Table 6 Mixture
proportions of grout CM
and EM from Zhang and
Benmokrane [30]
Type Mixture proportions
CM Type 10 SF cement (Blended Type I cement containing 8% silica fume) ? sand (50% by
weight of cement) ? superplasticizer solids (1.0% by weight of cement)
EM Type 10 SF cement (Blended Type I cement containing 8% silica fume) ? sand (50% by
weight of cement) ? superplasticizer solids (1.0% by weight of cement) ? swelling
agent (0.005% by weight of cement)
Table 7 Interfacial parameters obtained from Zhang and Benmokrane [30]
Types of specimens Corresponding specimens su (MPa) ss (MPa) d1 (mm) d2 (mm)
Round sanded ? EM Specimens SP1 and SP2 14.85 3.93 1.22 3.25
Ribbed ? CM Specimens SP3 and SP4 23.75 6.78 4.49 8.99
Ribbed ? EM Specimens SP5 and SP6 21.54 8.06 4.22 7.89
Materials and Structures (2010) 43:597–609 607
4 Conclusions
An analytical solution has been presented for pre-
dicting the maximum pull-out load of FRP rods
embedded in steel tubes filled with cement grout. In
the proposed solution, four parameters concerning the
rod–grout interface, i.e., the interfacial shear strength,
the slip corresponding to the shear strength, the
residual frictional stress and the slip when the
residual frictional stress first occurs, are needed.
The shear stress along the thickness direction of the
grout layer, the tensile stress in the rod and the
interfacial shear stress at the rod–grout interface have
been derived in an analytical manner. By modeling
interfacial debonding as an interfacial shear crack,
the pull-out load has been expressed as a function of
the interfacial crack length. With the help of the
Lagrange multiplier method, the maximum pull-out
load has been determined. By comparing the analyt-
ical solution with the experimental results obtained
from literature, it can be concluded that it can predict
the maximum pull-out load of spiral wound and
indented rods embedded in steel tubes filled with
cement grout with reasonable accuracy. But for the
rod with round sanded surface and low elastic
modulus, the proposed solution seems inapplicable
at the present stage. Besides, the proposed model can
be in principle extended to reinforced concrete, to
predict the pull-out capacity of a steel bar. It is
mainly because that the concrete Poisson’s ratio is
much smaller than that of the polymeric rods and the
shear-lag model is also applicable.
Acknowledgment The financial support from the National
Natural Science Foundation with Grant No. 50578025, of the
People’s Republic of China, is greatly acknowledged.
Open Access This article is distributed under the terms of the
Creative Commons Attribution Noncommercial License which
permits any noncommercial use, distribution, and reproduction
in any medium, provided the original author(s) and source are
credited.
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